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Introduction
The VSS power spectrum measurement, PWR_SPEC, produces a frequency domai=
n plot when performing time domain analysis. Because the power spectrum mea=
surement inherently performs a Fast Fourier Transform (FFT) operation, cert=
ain artifacts can arise in the data presented in the frequency domain. This=
article addresses the impact of FFT windowing type when using the power sp=
ectrum measurement. First, examples of FFT related phenomenon as witnessed =
in the power spectrum measurement will be shown. Next, some theory explaini=
ng these phenomenon will be presented. Finally, mitigation techniques to mi=
nimize some of the FFT nonideal effects will be discussed.
Nonideal FFT Effects
Most of the FFT effects on the power spectrum measurement discussed in t=
his article will concentrate on the FFT windowing type selected. FFT window=
type is selected as a secondary parameter in the PWR_SPEC measurement as s=
hown in Figures 1 and 2:
Figure 1. VSS PWR_SPEC Measurement.
Figure 2. Secondary Parameters for the P=
WR_SPEC Measurement.
The FFT Windowing defaults to Auto. Based on the kind =
of signal being measured, the PWR_SPEC measurement tries to select the best=
of the windowing types shown in the pick list. The window type selec=
ted can be viewed in the information box by left clicking on the trace in t=
he graph as shown in Figure 3:
Figure 3. PWR_SPEC Measurement Informati=
on Box
In most cases setting the FFT window type to Auto makes a good =
compromise, freeing the user from needing to consider an appropriate window=
type. However there will be situations where action is needed on selection=
of the FFT window type. Certain signal types and system configurations can=
lead to non desired power spectrum measurements. The next discussion highl=
ights some of the nonideal outcomes that may occur in the power spect=
rum measurement as result of FFT signal processing.
Resolving Closely Spaced Signals
Figure 4 shows three traces all making a PWR_SPEC measurement on the sam=
e system diagram. The only difference between the traces is the selection o=
f the windowing type.
Figure 4. Closely Spaced Signals with Di=
fferent FFT Windowing Types.
With the blue trace, the two CW signals are easy to resolve. However wit=
h the green trace and especially the red trace, the ability to resolve the =
two signals diminishes. As will be discussed later, different FFT windows e=
xpressed in the frequency domain yield different bandwidths. The wider the =
bandwidth, the lower the ability to resolve closely spaced signals.
Spectral Leakage
Figure 5 demonstrates spectral leakage that becomes apparent when the si=
gnal frequency is not exactly the same as one of the discrete FFT frequenci=
es:
Figure 5. Spectral Leakage in the PWR_SP=
EC Measurement.
For the left tone, the signal frequency is =E2=80=9Con bin=E2=80=9D, or =
exactly equal to one of the FFT frequencies, resulting in no spectral leaka=
ge. However for the tone on the right, its frequency is not on bin, resulti=
ng in an apparent spreading of the signal across the entire spectrum.
The two traces below show the power spectrum results from the same syste=
m diagram, the only difference being the selection of the windowing type.=
p>
Figure 6. Spectral Leakage with Differen=
t FFT Windowing Types
With proper selection of the FFT window, the amount of spectral leakage =
can be minimized. In this case the red trace=E2=80=99s spectral leakage is =
low enough to not mask the noise floor of the measurement.
=
Scalloping Loss
Shown in this graph are two signals whose amplitudes are identical. Howe=
ver in the measurement, there is an offset which in this case is 2.16 dB:=
p>
Figure 7. Scalloping Loss in the PWR_SPE=
C Measurement.
This amplitude offset results from the phenomenon known as scalloping lo=
ss which also occurs when the signal frequency is not equal to one of the d=
iscrete FFT frequencies.
Equivalent Noise Bandwidth
Another FFT window related anomaly occurs with noise and noise-like sign=
als such as digitally modulated signals. The following graph shows the spec=
trum from the same system diagram that has a CW signal and noise, the only =
difference between the two traces is the selection of the FFT window type:<=
/p>
Figure 8. Noise Floor Offset with Differ=
ent Windowing Types.
The amplitude of the CW signal for the two traces is the same, however t=
he noise floors show nearly a four dB offset from each other.
In this example, the amplitude of the digitally modulated signal shows a=
very similar offset as the noise floor example above.
Figure 9. Digitally Modulated Signal Powe=
r Offset with Different FFT Windowing Types.
Each FFT windowing type has a different noise power bandwidth. The FFT w=
indow expressed in the frequency domain exhibits a bandpass like filter abo=
ut each signal. The amount of noise power allowed to pass this bandpass dif=
fers with FFT window type resulting in an offset in the power spectrum meas=
urement for noise and digitally modulated signals.
Power Spectrum Measurement Bandwidth Types
Found in instrumentation such as spectrum analyzers are two bandwidth re=
lated parameters labelled Resolution Bandwidth Filter (RBW) and Video Bandw=
idth Filter (VBW). The VSS power spectrum measurement borrows these concept=
s using digital signal processing to realize these functions.
RBW in the Power Spectrum Measurement
The RBW serves as a bandpass filter centered on each signal in the spect=
rum. The narrower the bandwidth, the better the ability to separate closely=
spaced signals. The following plot shows a single tone with two different =
RBW settings:
Figure 10. RBW Setting Bandwidths
The red trace=E2=80=99s RBW setting is 10 times wider than the RBW setti=
ng used in the blue trace.
When two closely spaced signals are present, the two signals are only ap=
parent with the narrower RBW setting:
Figure 11. Resolving Closely Spaced Sign=
als with Different RBW Settings.
The other aspect of the RBW filter is the amount of noise allowed to pas=
s through the filter. The following shows the power spectrum with two diffe=
rent RBW settings:
<=
/p>
Figure 12. Noise Floor with Different RB=
W Settings.
The signal amplitude is the same for both RBW settings, however the nois=
e floor differs. The noise floor difference is equal to:
For this example with RBW settings that differ by a factor of 10, the no=
ise floor will differ by 10 dB.
However, just as in instrumentation, narrower RBW settings come at the p=
rice of measurement speed. The number of FFT points is directly proportiona=
l to the RBW setting. For instance in the above example, the 1 MHz RBW uses=
2000 FFT points, whereas the 100 kHz RBW uses 20,000 FFT points.
The number of FFT points can be determined using the information window:=
Figure 13. Information Box Showing the N=
umber of FFT Points
Often times the noise floor is specified in dBm/Hz, that is noise in a 1=
Hz bandwidth. When using a RBW other than 1 Hz, the 1 Hz noise power can b=
e determined from:
For instance if the RBW is 1 MHz, then the 1 Hz noise power can be found=
by subtracting 60 dB from the measured noise floor.
=
Auto FFT Bin Size
When selecting RBW by bandwidth value, the number of FFT points is limit=
ed to a maximum of 1,048,576. This limitation ensures that the simulation t=
ime is not excessive. However, this FFT size limitation can affect minimum =
RBW setting. When this occurs, then the number of FFT points will need to b=
e entered as the PWR_SPEC=E2=80=99s RBW/#Bins parameter.
This can be illustrated by example. The sampling frequency, as determine=
d from the trace=E2=80=99s information box, is 2 GHz. The user sets RBW to =
100 Hz. The number of FFT points that the PWR_SPEC measurement uses is comp=
uted from:
where Fs is the sampling frequency.
For this example, 2 GHz/100 Hz is 2e7 FFT points which exceeds the max F=
FT point limitation. When the maximum FFT point limitation is exceeded, the=
n the RBW is limited to
In this example, the information box shows the actual RBW even though th=
e user set the RBW parameter to 100 Hz:
Figure 14. Information Box Showing the Ac=
tual RBW when FFT Points are Limited to Default Maximum
Setting the PWR_SPEC measurement RBW/#Bins to 2e7 # FFT Bins, then the d=
esired 100 Hz RBW can be achieved.
Figure 15. Information Box Showing Correc=
ted RBW When Number of FFT Points are Entered Manually
Just be aware that this large a FFT size will make the simulation time d=
ramatically longer.
VBW in The Power Spectrum Measurement
The Video Bandwidth (VBW) filter is associated with trace averaging. The=
following plot shows the effect of trace averaging:
Figure 16. Noise Floor with Different Am=
ount of Trace Averaging.
Trace averaging is important when the signals of interest are close to t=
he noise floor.
The VSS power spectrum allows the selection of the VBW setting in either=
bandwidth or number of trace averages. If the bandwidth setting is chosen,=
the number of corresponding trace averages can be determined from the info=
rmation window that is accessed by clicking on the trace:
Figure 17. Information Box Showing Number=
of Trace Averages
Just as with instrumentation, more trace averages, or lower the VBW band=
width setting, the longer the measurement time.
DSP Sampling and Windowing Theory
This section provides a limited tutorial on the FFT windowing process th=
at is used in digital signal processing (DSP). For a more detailed explanat=
ion, consulting a textbook on the subject is recommended. The intent here i=
s to give enough of a theoretical understanding of the windowing process so=
that apparent anomalies viewed in the VSS power spectrum measurement resul=
ts can be better explained. With this understanding will come mitigation te=
chniques to achieve results that closer match expected results.
=
Sampling Theory
A well known property of the Fourier transform is that a pulse in the ti=
me domain results in a sinc function, sin(x)/x, in the frequency domain. Sh=
own below is the magnitude expressed in dB of the frequency domain sinc fun=
ction:
Figure 18. Log Magnitude of the Sinc Fun=
ction in the Frequency Domain
This importance of this fundamental concept becomes apparent when signal=
sampling is considered.
At the heart of digital signal process is the sampling of continuous sig=
nals as shown below:
Figure 19. Sampling of a Continuous Time=
Domain Signal.
The ideal sampler functions as a switch closing once every sampling peri=
od, Ts. The sampling frequency, Fs, is 1/Ts. The result is the sampled sign=
al with finite sampling duration. The time domain appears as a pulsed RF an=
d it is the pulse in the time domain that gets transformed into the sinc re=
sponse in the frequency domain. Expressed in the frequency domain, the CW s=
ignal shows the sinc function superimposed upon it:
Figure 20. FFT Spectrum of a CW Signal wh=
ose Frequency Matches One of the FFT Frequencies.
The bins in the frequency domain are separated by Fs/N where N is the nu=
mber of time domain samples. The literature mentions the use of zero stuffi=
ng to decrease the frequency bin spacing; the assumption here is that zero =
stuffing is not used.
As depicted above, ideally the sinc nulls fall on the frequency bins, re=
sulting in a single spectral response located at the signal frequency. But =
this only occurs if the signal frequency falls on an FFT frequency bin, tha=
t is signal frequency is:
Spectral Leakage
In general, the signal frequency is not an integral submultiple of Fs/N.=
The following depicts the situation where the signal frequency is not on b=
in:
Figure 21. Sampling and FFT of a CW Signa=
l where the Frequency does not Match One of the FFT Frequencies.
In the time domain at the end of the sampling interval, there is an abru=
pt transition. In the frequency domain, the sinc pulse that is superimposed=
upon the signal frequency spectral component shows that the nulls of the s=
inc response do not fall exactly on the frequency bins. There is spectral e=
nergy at multiple bins. This phenomenon is known as spectral leakage.
The spectral leakage becomes a problem in that the leakage from one larg=
er amplitude signal can mask a nearby lower amplitude signal as depicted he=
re:
Figure 22. Spectral Leakage Masking a Nea=
rby Lower Amplitude Signal
The next section will address how spectral leakage can be suppressed.
FFT Windowing
Suppressing spectral leakage is accomplished by applying a FFT window to=
the time domain samples before performing the FFT operation. The idea is t=
o taper the time domain waveform at the start and end of the sampling inter=
val so that abrupt edges are minimized.
Figure 23. Sampling with FFT Windowing F=
unction Applied
The following plot shows time domain responses of a CW signal where the =
lighter trace shows the unwindowed time domain waveform of the signal and t=
he darker traces shows the windowed waveform:
Figure 24. Windowing in the Time Domain<=
/p>
In the frequency domain, the spectral leakage is dramatically minimized =
when FFT windowing is applied. Here the red trace is the frequency domain s=
pectrum without windowing and the blue is the same time domain sample, but =
with windowing applied:
Figure 25. Spectral Leakage with and wit=
hout Windowing Applied.
The signal at 0.5 GHz that was completely masked by the spectral leakage=
of the unwindowed spectrum is now clearly visible once windowing is applie=
d.
FFT Window Properties
With the general theory of FFT windowing introduced, this section review=
some of the windowing properties that distinguish one window type from ano=
ther.
In the VSS power spectrum measurement, there are several FFT windowing t=
ypes in which to choose. Selecting the proper window for any given measurem=
ent scenario requires understanding the properties of the windowing functio=
n. The following graphs in this section show time domain and frequency doma=
in comparisons of a few different windowing functions. Each signal response=
uses 4096 samples.
Time Domain Envelope
Most windowing functions taper the time domain waveform such that the wa=
veform amplitudes are zero at the start and stop of the sampling interval. =
But some windowing functions, such as the Kaiser window shown here, leave a=
small discontinuity at the sampling interval edges:
Figure 26. FFT Window Time Domain Envelo=
pes
The Rectangular widow has a very abrupt edge. It may not seem appropriat=
e, but no windowing is considered a windowing function. The selection in th=
e VSS power spectrum measurement this is labelled None and in the =
literature this is labelled Rectangular. Later in this discussion =
it will become apparent that this abrupt edge has some serious performance =
limitations.
Although somewhat interesting, the time domain envelopes themselves do n=
ot give much useful information on the frequency domain properties of the w=
indow. Viewing the windowing function in the frequency domain is most appro=
priate for the power spectrum measurement.
3 dB Bandwidth
The 3 dB bandwidth is considered an important figure of merit for determ=
ining the ability to separate closely spaced signals. Shown here are the fr=
equency domain responses of a few windows:
Figure 27. FFT Windowing 3 dB Bandwidths=
The 3 dB bandwidth expressed in the number of FFT frequency bins ranges =
from 0.88 for the Rectangular window to 1.44 for the Hanning (the names Han=
n and Hanning are both used interchangeably in the literature) window.
For the VSS power spectrum measurement, the parameter RBW/# Bins is used=
to set the frequency spacing. Independent of FFT windowing type selected, =
the frequency spacing between frequency bins is always Fs/N where N is the =
number of FFT points. If the user sets RBW/# Bins parameter to a bandwidth =
value, then the frequency spacing will be that bandwidth value. The number =
of FFT points will be adjusted accordingly. For instance if the RBW/# Bins =
parameter is set to 1 MHz, then the frequency spacing of the power spectrum=
measurement will be 1 MHz. This behavior is different from most instrument=
ation where the RBW setting often implies the 3 dB bandwidth of the resolut=
ion bandwidth filter. This is an important distinction between VSS and inst=
rumentation and can lead to confusion if not understood.
Because VSS power spectrum measurement keeps a constant frequency bin sp=
acing independent of FFT windowing type selected, the resolving power of cl=
osely spaced signals is dependent of which windowing type has been selected=
. The windowing type with the lower 3 dB bin value will have a better abili=
ty to resolve closely spaced signals. The windowing type available in the P=
WR_SPEC measurement with the narrowest 3 dB bandwidth expressed in frequenc=
y bins is the Rectangular or None window.
Equivalent Noise Bandwidth
It is readily apparent by observing the FFT windowing function responses=
in the frequency domain in Figure 27 that each windowing function has a di=
fferent frequency response. Integrating the area under the magnitude square=
d of the windowing frequency response yields the windowing function=E2=80=
=99s power gain. Multiplying the windowing function=E2=80=99s power gain by=
the noise power results in the windowing function=E2=80=99s noise power. B=
ecause of the varying frequency responses of each windowing functions, the =
noise power will also vary between windowing functions.
A figure of merit is the Equivalent Noise Bandwidth (ENBW). If =
a perfect brick-wall filter could be constructed such that its power gain i=
s the same as the power gain of the windowing function, the bandwidth of th=
e brick-wall filter would be ENBW as depicted in Figure 28:
Figure 28. Equivalent Noise Bandwidth
ENBW is wider than the 3 dB bandwidth and it is also wider than the freq=
uency bin spacing. The VSS power spectrum measurement uses the same frequen=
cy bin spacing independent of the FFT windowing function selected and as a =
result the noise power will vary with FFT windowing function. This behavior=
is shown in Figure 29:
Figure 29. Noise Power Differences with =
Different FFT Windowing Functions.
The noise power is predictable and is based on ENBW expressed in number =
of frequency bins. For the Rectangular window, the ENBW is 1 bin wide. For =
the Flat Top window, the ENBW is 3.77 bins wide. The noise offset is given =
by:
For the Flat Top windowing function the noise power offset is 5.76 dB. F=
or the Rectangular window, the noise power offset is 0 dB (i.e. no offset).=
Sidelobe Suppression
Another very important aspect of the windowing function relates to the s=
uppression of spectral leakage. The literature refers to this figure of mer=
it as Sidelobe Suppression. Figure 30 shows the sidelobe suppressi=
on of a few windowing functions over a relative narrow frequency range.
Figure 30. Narrowband View of FFT Window=
ing Sidelobes.
In figure 31 the same windowing functions are shown over a wider frequen=
cy range.
Figure 31. Wideband View of FFT Windowin=
g Sidelobes.
Note: only the peaks of the sidelobes are plotted in Figure 31. Maximum =
sidelobe suppression occurs at a frequency offset that is 25% of the number=
of FFT points. As Figure 31 demonstrates, the selection of the windowing f=
unction has a huge impact on the amount of spectral leakage.
Scalloping Loss
As mentioned in the discussion centered around Figure 7, amplitudes of d=
isplayed CW signals can exhibit an offset as result of a phenomenon known a=
s Scalloping Loss. Scalloping loss occurs when the signal frequenc=
y does not fall on a FFT frequency bin. As explained for the reason behind =
spectral leakage, the windowing function displayed in the frequency domain =
is centered on the signal. However, spectral energy can only be displayed a=
t the bin frequencies. So, the amplitude of the signal at the closest bin f=
requency to the signal frequency will be the highest amplitude, but as illu=
strated in Figure 32, that amplitude will be attenuated due to the superimp=
osed windowing response:
Figure 32. Scalloping Loss
The scalloping loss is greatest when the signal frequency falls half way=
between FFT frequency bins. And because the windowing function frequency r=
esponse differs between windowing types, the amount of scalloping loss will=
also differ with windowing type. Figure 33 shows the scalloping loss (dark=
er trace) using the Rectangular windowing function.
Figure 33. Rectangular Window Scalloping=
Loss
For the Rectangular window, the maximum scalloping loss is nearly 4 dB. =
In Figure 34 where the 4-term Blackmann-Harris window is shown, the scallop=
ing loss is considerably lower at less than 1 dB.
Figure 34. 4-term Blackmann-Harris Windo=
w Scalloping Loss
Comparison of FFT Windowing Function Performan=
ce
Above the reasoning behind the various windowing figures of merit was pr=
esented. This section now summarizes the figures of merits for the windowin=
g functions that are used by VSS.
The following table shows figures of merit for 3 dB BW, ENBW and scallop=
ing loss for the windowing functions available in the VSS power spectrum me=
asurement.
Table 1. FFT Windowing Function 3 dB BW,=
ENBW, and Scalloping Loss
Sidelobe suppression for these windowing types is shown in Figure 35 and=
Table 2.
Figure 35. Sidelobe Frequency Response=
p>
Table 2. Tabular Results for Sidel=
obe Suppression.
This data shows that a large selection of windowing functions exist beca=
use the performance of each varies so much. Unfortunately each one has a tr=
ade off. For instance, for best amplitude accuracy the Flat Top is the best=
choice due to its low scalloping loss. However, Flat Top has the widest ba=
ndwidth, so resolving closely spaced signals is problematic and Flat Top ha=
s the highest noise power offset. Rectangular is the best choice for resolv=
ing power, but the worst choice for spectral leakage.
Which FFT Windowing Function to Use?
The question often comes up, what is the recommended windowing function?=
The quick answer is that each system is unique and it is impossible to mak=
e a general recommendation. With the information provided in the above tabl=
es, the user now has the information to assess their particular situation a=
nd choose the windowing function accordingly.
Fortunately VSS tries to assess the system and choose the best windowing=
function if the windowing type is set to the Auto position, which=
is the default. System assessment is based largely on the signal type (mod=
ulated vs. CW) and number of trace averages among other criterion.
Compensating for ENBW
Scalloping loss and spectral leakage can not be compensated. The only fa=
llback is to choose another windowing type to affect the behavior. However,=
the noise floor offset is something that can be compensated. A device know=
as processing gain can be used to adjust the displayed power of n=
oise and noise-like signals. This property can be used to compensate for th=
e noise floor and digitally modulated signal power offset.
=
Processing Gain
Figure 12 demonstrates that the noise floor changes with RBW setting. Be=
cause RBW is a function of the number of FFT points, then it follows that t=
he noise floor is a function of the number of FFT points. This is known in =
the literature as processing gain. The amount of processing gain o=
f noise and noise-like signals is given by:
where N1 and N2 are the number of FFT points. So b=
y doubling the number of FFT points, the noise floor drops 3.01 dB. By incr=
easing the number of FFT points by a factor of 10 drops the noise floor by =
10 dB.
Using Processing Gain in the PWR_SPEC Measu=
rement
In the PWR_SPEC measurement, the number of FFT Bins can be used instead =
of RBW setting as shown in Figure 36:
Figure 36. Entering Number of FFT Bins i=
n the PWR_SPEC Measurement.
For all windowing types, the displayed noise power offset is an increase=
. So the compensation required is to decrease the noise power offset. First=
set the RBW to a desired bandwidth and simulate. View the number of FFT bi=
ns for that measurement using the information box (left click on the trace =
in the graph). Use Table 1 to look up the ENBW bins for the selected window=
ing type. Multiply the number of bins by the ENBW bins. Set the #FFT Bins i=
n the PWR_SPEC measurement for the calculated number of bins.
Figure 37 shows an example using the Hann window. The red trace shows th=
e response with RBW set to 1 MHz. The number of FFT bins for this measureme=
nt is 2000. Table 1 shows that the ENBW bins for the Hann window is 1.5. So=
the new number of FFT bins is 2000 x 1.5 =3D 3000. Set #FFT Bins in the PW=
R_SPEC measurement to 3000 and the result is the blue trace.
Figure 37. Compensated Noise Floor
Note the marker value of the blue trace being -114 dBm. This is measured=
in a 1 MHz RBW that has been compensated for ENBW. To get the 1 Hz noise p=
ower density, subtract 10*log10(1 MHz) =3D 60 dB and the result =
is the correct value of -174 dBm/Hz.
Use Noise Offset from Figure of Merit Table
The alternative that does not require adjusting the number of FFT bins i=
s to use Table 1, Noise Power Offset due to ENBW[dB] column. Use t=
he values in that column to subtract the offset from the measured noise pow=
er or the power of noise-like signals.
Use the PWR_MTR Measurement
For accurate power of noise and noise-like signals another alternative i=
s to use the PWR_MTR measurement and adjust the Band Center and Bandwidth p=
arameters appropriately.
Figure 38. PWR_MTR Measurement
The PWR_MTR measurement compensates for noise power offset so that it re=
ports accurate noise power for all FFT Window type settings.
Conc=
lusion
For the RF designer using VSS for system level analysis, certain aspects=
of using time domain analysis may come as a surprise. Many of the seemingl=
y incorrect simulation results stem from the FFT windowing process. This ar=
ticle has provided some back ground on the windowing process so that now th=
e user can make the necessary adjustments to compensate for these nonideal =
outcomes.
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